# Factors and Multiples

Factors and multiples are **different** things.

But they both involve **multiplication**:

- Factors are what we can multiply to get the number
- Multiples are what we get
**after**multiplying the number by an integer (not a fraction).

### Example: the positive factors, and some multiples, of 6:

Factors:

- 1 × 6 = 6, so
**1**and**6**are factors of 6 - 2 × 3 = 6, so
**2**and**3**are factors of 6

Multiples:

- 0 × 6 = 0, so
**0**is a multiple of 6 - 1 × 6 = 6, so
**6**is a multiple of 6 - 2 × 6 = 12, so
**12**is a multiple of 6 - and so on

(Note: there are negative factors and multiples as well)

Here are the details:

## Factors

"Factors" are the numbers we can **multiply together** to get another number:

2 and 3 are factors of 6

A number can have **many** factors.

### Example: 12

- 3 × 4 = 12, so
**3**and**4**are factors of 12 - Also 2 × 6 = 12, so
**2**and**6**are also factors of 12, - And 1 × 12 = 12, so
**1**and**12**are factors of 12 as well.

AND because multiplying negatives makes a positive, −1, −2, −3, −4, −6 and −12 are also factors of 12:

- (−1) × (−12) = 12
- (−2) × (−6) = 12
- (−3) × (−4) = 12

So ALL the factors of 12 are:

**1, 2, 3, 4, 6 and 12**

AND **−1, −2, −3, −4, −6 and −12**

Learn about Greatest Common Factor and how to find All Factors of a Number.

## Multiples

A multiple is the result of **multiplying** a number **by an integer** (not a fraction).

### Example: Multiples of 3 are

..., −9, −6, −3, 0, 3, 6, 9, ...

Example: 15 **is** a multiple of 3, as 3 × 5 = 15

Example: 16 is **not** a multiple of 3

### Example: Multiples of 5 are

..., −15, −10, −5, 0, 5, 10, 15, ...

Example: 10 **is** a multiple of 5, as 5 × 2 = 10

Example: 11 is **not** a multiple of 5

### Multiples of Anything

We must **multiply by an integer**, but the number that is being multiplied can be anything.

### Example: Multiples of π

..., −2π, −π, 0, π, 2π, 3π, 4π, ...